{"paper":{"title":"Extremal Theta-free planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yongtang Shi, Yongxin Lan, Zi-Xia Song","submitted_at":"2017-11-05T16:50:52Z","abstract_excerpt":"Given a family $\\mathcal{F}$, a graph is $\\mathcal{F}$-free if it does not contain any graph in $\\mathcal{F}$ as a subgraph. We study the topic of \"extremal\" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213--230], that is, how many edges can an $\\mathcal{F}$-free planar graph on $n$ vertices have? We define $ex_{_\\mathcal{P}}(n,\\mathcal{F})$ to be the maximum number of edges in an $\\mathcal{F}$-free planar graph on $n $ vertices. Dowden obtained the tight bounds $ex_{_\\mathcal{P}}(n,C_4)\\leq15(n-2)/7$ for all $n\\geq4$ and $ex_{_\\mathcal{P}}(n,C_5)\\leq(12n-33)/5$ for all $n\\geq1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.01614","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}